Breathing helium is actually rather dangerous, as it can lead to asphyxiation because your body receives much less oxygen. When you sing, you’re not only producing the frequency of one note, but also adding many more frequencies that make your voice unique. The empty space around your throat and mouth acts like a resonance box in a musical instrument. Smaller boxes favour higher frequencies or pitches, while larger ones do the same for lower frequencies.
Sound travels much faster through helium than it does through the air. As sound can travel back and forth much faster if those cavities are filled with helium, the effect is like having a smaller box, and so higher frequencies will be favoured.
José Monteiro, Science Museum